Convex duality and intertemporal consumption choice Peter Bank and - PowerPoint PPT Presentation

Convex duality and intertemporal consumption choice Peter Bank and joint work with Helena Kauppila, Ph.D. 2009 at Columbia University AnStAp10 Conference in Honor of W. Schachermayer Vienna, July 12–16, 2010

Utility maximization How to optimally invest in a financial market? ◮ financial market: tradable assets S with equivalent local martingale measures M � = ∅ , S 0 ≡ 1 � t ◮ trade: X t = x + 0 H s dS s ≥ 0 ❀ X ∈ X ( x ) ◮ primal problem: Maximize E U ( X T ) with U strictly increasing, concave, U ′ ( 0 ) = ∞ , U ′ ( ∞ ) = 0 ◮ u ( x ) � sup X ∈ X ( x ) E U ( X T ) < ∞

Utility maximization How to optimally invest in a financial market? ◮ financial market: tradable assets S with equivalent local martingale measures M � = ∅ , S 0 ≡ 1 � t ◮ trade: X t = x + 0 H s dS s ≥ 0 ❀ X ∈ X ( x ) ◮ primal problem: Maximize E U ( X T ) with U strictly increasing, concave, U ′ ( 0 ) = ∞ , U ′ ( ∞ ) = 0 ◮ u ( x ) � sup X ∈ X ( x ) E U ( X T ) < ∞ Kramkov-Schachermayer (1999, 2003): U ′ ( x ) ◮ AE ( U ) = lim x →∞ U ( x ) / x < 1 ◮ bipolar theorem generalizing financiability constraint X ≤ X T ∈ X ( x ) ⇔ sup Q ∈ M E Q X ≤ x ◮ convex duality ◮ . . .

Convex duality à la Kramkov-Schachermayer ◮ Legendre-Fenchel transform: V ( y ) = sup x ≥ 0 { U ( x ) − xy } ◮ dual processes: Y s.t. Y 0 = y , XY supermartingale for all X ∈ X ( 1 ) ❀ Y ∈ Y ( y ) ◮ dual problem: Minimize E V ( Y T ) over Y ∈ Y ( y ) ◮ v ( y ) = inf Y ∈ Y ( y ) E V ( Y T )

Convex duality à la Kramkov-Schachermayer ◮ Legendre-Fenchel transform: V ( y ) = sup x ≥ 0 { U ( x ) − xy } ◮ dual processes: Y s.t. Y 0 = y , XY supermartingale for all X ∈ X ( 1 ) ❀ Y ∈ Y ( y ) ◮ dual problem: Minimize E V ( Y T ) over Y ∈ Y ( y ) ◮ v ( y ) = inf Y ∈ Y ( y ) E V ( Y T ) Theorem (Kramkov-Schachermayer) If v ( y ) < ∞ for all y > 0 : 1. u and v are C 1 , strictly convex, and conjugate to each other: v ( y ) = sup { u ( x ) − xy } , u ( x ) = inf y ≥ 0 { v ( y ) + xy } x ≥ 0 2. For y = u ′ ( x ) , the optimizers for primal and dual problem exists and are related by U ′ ( X ∗ T ( x )) = Y ∗ X ∗ T ( x ) = − V ′ ( Y ∗ T ( y ) , T ( y )) .

Convex duality and intertemporal consumption Our aim: In analogy to Kramkov-Schachermayer, develop a theory of convex duality for utility functionals of the form � T E U ( C ) = E F ( t , C t ) dt 0 where ◮ F = F ( t , c ) is an agent’s felicity function : strictly concave and increasing in c , Inada conditions, AE ( F )( t , c ) < 1 ◮ C = ( C t ) 0 ≤ t ≤ T is the agent’s cumulative consumption : ≥ 0, increasing, adapted, left-continuous

Why should one consider such a utility functional? � T E U ( C ) = E F ( t , C t ) dt 0 ◮ F = F ( t , c ) is an agent’s felicity function : C 1 , 2 , strictly concave and increasing in c , Inada conditions, AE ( F )( t , c ) < 1 ◮ C = ( C t ) 0 ≤ t ≤ T is the agent’s cumulative consumption : ≥ 0, increasing, adapted, left-continuous Different answers from different people, e.g., from a . . .

Why should one consider such a utility functional? � T E U ( C ) = E F ( t , C t ) dt 0 ◮ F = F ( t , c ) is an agent’s felicity function : C 1 , 2 , strictly concave and increasing in c , Inada conditions, AE ( F )( t , c ) < 1 ◮ C = ( C t ) 0 ≤ t ≤ T is the agent’s cumulative consumption : ≥ 0, increasing, adapted, left-continuous Different answers from different people, e.g., from a . . . mountaineer: Because it’s there!

Why should one consider such a utility functional? � T E U ( C ) = E F ( t , C t ) dt 0 ◮ F = F ( t , c ) is an agent’s felicity function : C 1 , 2 , strictly concave and increasing in c , Inada conditions, AE ( F )( t , c ) < 1 ◮ C = ( C t ) 0 ≤ t ≤ T is the agent’s cumulative consumption : ≥ 0, increasing, adapted, left-continuous Different answers from different people, e.g., from a . . . mountaineer: Because it’s there! agnostic: Because it leaves open whether optimal consumption is done at rates, in gulps or other ways.

Why should one consider such a utility functional? � T E U ( C ) = E F ( t , C t ) dt 0 ◮ F = F ( t , c ) is an agent’s felicity function : C 1 , 2 , strictly concave and increasing in c , Inada conditions, AE ( F )( t , c ) < 1 ◮ C = ( C t ) 0 ≤ t ≤ T is the agent’s cumulative consumption : ≥ 0, increasing, adapted, left-continuous Different answers from different people, e.g., from a . . . mountaineer: Because it’s there! agnostic: Because it leaves open whether optimal consumption is done at rates, in gulps or other ways. aesthete: Because the standard setting of felicity from consumption rates, i.e., speed of consumption does not exactly reflect savoir vivre .

Why should one consider such a utility functional? � T E U ( C ) = E F ( t , C t ) dt 0 ◮ F = F ( t , c ) is an agent’s felicity function : C 1 , 2 , strictly concave and increasing in c , Inada conditions, AE ( F )( t , c ) < 1 ◮ C = ( C t ) 0 ≤ t ≤ T is the agent’s cumulative consumption : ≥ 0, increasing, adapted, left-continuous Different answers from different people, e.g., from a . . . mountaineer: Because it’s there! agnostic: Because it leaves open whether optimal consumption is done at rates, in gulps or other ways. aesthete: Because the standard setting of felicity from consumption rates, i.e., speed of consumption does not exactly reflect savoir vivre . economist: Because it allows for intertemporal substitution of consumption; see Hindy, Huang, Kreps (1992, 1993)

Problem formulation Utility maximization Maximize E U ( C ) over all consumption plans C which the agent can finance by investing the initial wealth x > 0 in the stock market: u ( x ) = sup E U ( C ) C ∈ C ( x ) where � C ( x ) = { C ≥ 0 , ↑ , l.c. , adapted, E Y t dC t ≤ x , Y ∈ Y ( 1 ) } [ 0 , T ) Benth et al. (2001): exponential Levy model, HARA-felicity, HJB-approach Our view: Maximize concave functional E U ( C ) subject to linear constraints.

The first step: Compute the Legendre-Fenchel Transform For deterministic Y = ( Y t ) ≥ 0, consider the functional � � � V ( Y ) = sup U ( C ) − Y t dC t ( ∗ ) dC ≥ 0 [ 0 , T )

The first step: Compute the Legendre-Fenchel Transform For deterministic Y = ( Y t ) ≥ 0, consider the functional � � � V ( Y ) = sup U ( C ) − Y t dC t ( ∗ ) dC ≥ 0 [ 0 , T ) If Y > 0 and l.s.c., solution C ∗ can be computed from time-inhomogeneous convex envelope � Y ∗ ≤ Y s ) ds , dC ∗ ≥ 0 Y ∗ F ′ ( s , C ∗ maximal with t = [ t , T ) where F ′ ( t , c ) = ∂ ∂ c F ( t , c ) . In particular: � � � U ( C ) = inf V ( Y ) + Y t dC t Y ≥ 0 [ 0 , T )

Convex duality in the deterministic case Corollary For Y ∗ and C ∗ as above, the following statements are equivalent: � � (i) C ∗ attains sup dC ≥ 0 � [ 0 , T ) Y ∗ U ( C ) − t dC t (ii) Y ∗ attains inf Y ≥ 0 � � � [ 0 , T ) Y t dC ∗ V ( Y ) + t (iii) ∇ U ( C ∗ ) = inhomogeneously convex envelope of Y ∗ 1 [ 0 , T ) (iv) ∇ V ( Y ∗ ) = − dC ∗

Convex duality in the stochastic case For Y ∈ Y ( y ) , consider the transform � � � sup E U ( C ) − Y t dC t ( ∗∗ ) dC ≥ 0 , adapted [ 0 , T ) Duality gap? In general � � �� � ( ∗∗ ) < E V ( Y ) = E sup U ( C ) − Y t dC t dC ≥ 0 [ 0 , T ) because ω -wise argmax will not give an adapted consumption plan, even if Y is adapted: non-time additive structure of U ❀ we need to relax our choice of Y to address adaptivity constraint; cf. Davis, Karatzas (1994)

Convex duality in the stochastic case Well known: � � ˜ E Y t dC t = E Y t dC t for all C [ 0 , T ) [ 0 , T ) iff E Y τ = E ˜ Y τ for all stopping times τ < T : optional projection

Convex duality in the stochastic case Well known: � � ˜ E Y t dC t = E Y t dC t for all C [ 0 , T ) [ 0 , T ) iff E Y τ = E ˜ Y τ for all stopping times τ < T : optional projection Should we thus enlarge the space of dual variables to all ˜ Y with optional projection o ˜ Y = Y : � � � E V ( ˜ sup U ( C ) − Y t dC t = inf Y ) ? E o ˜ dC ≥ 0 , adapted [ 0 , T ) Y = Y

Convex duality in the stochastic case Well known: � � ˜ E Y t dC t = E Y t dC t for all C [ 0 , T ) [ 0 , T ) iff E Y τ = E ˜ Y τ for all stopping times τ < T : optional projection Should we thus enlarge the space of dual variables to all ˜ Y with optional projection o ˜ Y = Y : � � � E V ( ˜ sup U ( C ) − Y t dC t = inf Y ) ? E o ˜ dC ≥ 0 , adapted [ 0 , T ) Y = Y Unclear how to construct optimal C ∗ from minimzer ˜ Y ∗ of dual problem: non-uniqueness, adaptedness of argmax?

Convex duality in the stochastic case Idea: confine dual space to ‘convex’ processes For Y ∈ Y ( y ) find ˜ Y ≤ Y with inhomgeneously convex paths and adapted density ∂ t ˜ Y t = − F ′ ( t , C t ) such that o ˜ Y t = Y t whenever dC t > 0 . B., El Karoui (2004): Such a probabilistic analog of an inhomogeneously convex envelope for Y can be found in exactly one way.